Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{k^3 + 14k^2 + 49k}{k^3 - 49k}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {k(k^2 + 14k + 49)} {k(k^2 - 49)} $ $ x = \dfrac{k}{k} \cdot \dfrac{k^2 + 14k + 49}{k^2 - 49} $ Simplify: $ x = \dfrac{k^2 + 14k + 49}{k^2 - 49}$ Since we are dividing by $k$ , we must remember that $k \neq 0$ Next factor the numerator and denominator. $ x = \dfrac{(k + 7)(k + 7)}{(k + 7)(k - 7)}$ Assuming $k \neq -7$ , we can cancel the $k + 7$ $ x = \dfrac{k + 7}{k - 7}$ Therefore: $ x = \dfrac{ k + 7 }{ k - 7 }$, $k \neq -7$, $k \neq 0$